Toy having lissajous vibratory motion



June 23, 1970 ENGELMAN 3,5l,193

TOY HAVING LISSAJOUS VIBRATORY MOTION Filed 001;. 11, 1968 2Sheets-Sheet 1 I4 A l A FIG.....4

' INVENTOR.

D ONALD MAX ENGELMAN FIG 3 WT ATTORNEYS JUIIE 23, D, M, EN 3,516,193

TOY HAVING LISSAJOUS VIBRATORY MOTION Filed Oct. 11, 1968 2 Sheets-SheetB INVENTOR. DONALD MAX ENGELMAN ATTORNEYS United States Patent 3,516,193TOY HAVING LISSAJOUS VIBRATORY MOTION Donald Max Engelman, Palo Alto,Calif., assignor to Kinetic Objects Inc., San Anselmo, Calif. Filed Oct.11, 1968, Ser. No. 766,896 Int. Cl. A63h 33/00 US. Cl. 46-1 ClaimsABSTRACT OF THE DISCLOSURE A toy including a mass mounted on thenonsupported end of an elastic centilever for vibrational movement. Thecantilever, typically constructed from a piece of wire, has a pluralityof coplanar zigzags disposing segments of the wire substantially normalto the cantilever length." These segments provide substantiallyindependent torsion and bending forces acting at right angles one to theother. The mass, when displaced and set in motion at the nonsupportedend of a cantilever, undergoes vibrational motion deflected by theindependent restoring forces and consequently traces Lassajous figuresalong its vibratory path.

This invention relates to a vibrating toy and more particularly to acantilever supported mass which traces Lissajous figures along itsvibratory path.

Mechanical devices which trace Lissajous figures along their vibratorypath have heretofore been known. These devices, however, have comprisedcomplex pendulums or other mechanisms, which have included specializedhinges and drives to provide the substantially independent sinusoidalforces necessary for the generation of the Lissajous figures.

In contradistinction to these heretofore known devices, this inventionprovides a simple vibratory mechanism capable of readily tracingLissajous figures along its vibratory path. Accordingly, an elasticcantilever is provided having a mass mounted for vibrational motion atthe free or nonsupported end. The elastic cantilever has providedtherein substantially independent restoring forces, each restoring forceacting along an axis intersecting the axis of the other restoring force.When the mass is displaced and set in motion at an acute or obtuse angleto the axes of the cantilevers restoring forces, it is subjected to thesimultaneous interaction of both restoring forces, tracing Lissajousfigures along its vibratory path.

A further object of this invention is to provide a cantileverconstructed from a piece of elastic wire which has substantialyindependent restoring forces along mutually perpendicular axes.According to this aspect a wire having differing elastic restoringforces in bending and torsion is provided with a plurality of zigzagsegments. These zigzag segments, are all disposed substantialyhorizontaly to the longitudinal axis of the cantilever within a commonplane including the longitudinal axis of the cantilever. When themounted mass vibrates within the plane of the zigzag, the horizontalsegments of the wire are subjected to pure bending forces, giving thevibrating mass a first sinusoidal motion. When the mounted mass isvibrated normally to the plane of the zigzag, the horizontal seg mentsof the wire are subjected substantially to pure torsion forces, givingthe vibratory mass a second sinusoidal motion normal to the firstsinusoidal motion. By displacing or setting the mass in motion at acuteor obtuse angles from the common plane of the zigzag segments, the massvibrates along a path which traces Lissajous fiigures.

A further object of this invention is to provide the mass undergoingvibration with a vibratory frequency wherein the Lissajous pattern ofits vibration can be readily observed. Accordingly, the mass andcantilever are preselected to provide constant vibratory oscilations atnear but not equal rates along each of the intersecting axes. These nearbut not equal rates are each in a range between 10-100 cycles perminute. When the mass is displaced and set in motion, it traces aLissajous pattern which can be plainly seen.

A further object of this invention is to provide a cantileverconstructed of wire, in which the zigzag bends impart maximum dependenceof their respective independent restoring forces in bending and torsion.Accordingly, the elastic wire cantilever has its horizontal zigzagsegments close to the supported end. When the vibrating mass is set inmotion, maximum elastic deflection of the wire cantilever occursadjoining its supported end, accentuating the substantially independentrestoring forces of the wire and emphasising the Lissajous vibratorypath of the mass.

Oher objects, features and advantages of the present invention will bemore apparent after referring to the following specification andattached drawings in which:

FIG. 1 is a front elevation of the vibratory toy showing its lowerzigzag segments in the plane of the figure;

FIG. 2 is a side elevation of the vibratory toy taken at right angles tothe view of FIG. 1 showing the upper zigzag segment in the plane of thefigure;

FIG. 3 is a side elevation of the vibratory toy vibrating so that itslower zigzag segments are in the stable or torsion mode;

FIG. 4 is a plan view of the vibratory toy vibrating so that the lowerzigzag segments are in the unstable or bending mode;

FIG. 5a through 5d show plan views of the vibratory toy undergoingsequential phases of vibratory movement tracing Lissajous figures inwhich:

5a shows a substantially linear phase of vibratory motion of the mass ata small acute angle counterclockwise from the plane of the lower zigzagsegments.

5b shows a substantially circular phase of vibratory motion of the massin a counterclockwise direction about the cantilever base,

50 shows a second substantially linear phase of vibratory motion of themass at a small acute angle clockwise from the plane of the lower zigzagsegments, and

5d shows a substantially circular phase of vibratory motion of the massin a clockwise direction about the cantilever base; and

FIG. 6 illustrates an alternate cross-sectional construction of acantilever useful for the practice of this invention.

With reference to the figures, the toy comprises a support A havingelastic cantilever C embedded therein with ball or mass B attached tothe free end of the cantilever for vibratory movement about the support.

Support A can be constructed of a hardwood block of approximately 400grams of weight with a base 4-5 inches square. This base has a width andweight sufficient to hold the cantilever and ball stationary while theyundergo vibratory motion. Ball B is a plastic mass weighingapproximately /8 that of the base or 45 grams.

Cantilever C comprises inch tinned music wire bent so as to have aplurality of coplanar zigzag segments disposed normally to itslongitudinal axis. Cantilever C is embedded within and supported bysupport A and extends vertically upward suspending mass B at itsnonsupported upward extremity. Between support A and mass B cantilever Cis here shown having a first set of planar zigzags 14 disposed in thecantilever adjacent support A and a second set of planar zigzags 16disposed adjacent mass B in cantilever C.

Describing the zigzags, zigzags 14 comprise substantially horizontalcantilever segments 20, 21 and 22. These segments have located betweentheir lengths opposed 1cute bends 24 and 25, bend 24 being betweensegnents and 21 and bend 25 being between segments 21 and 22. A shortvertical segment 27 extends between the lower portion of segment 20 andbase A. As is apqarent from the view of FIG. 2, segments 20-22, are 11].disposed in coplanar relation along a plane includng the vertical axisof cantilever C.

Similar to first planar zigzags 14, second planar zigtags 16 includesubstantially horizontal segments 30, 31, and 32. Opposed bends 34 and35 interconnect segnents 30 and 31, and segments 31 and 32, respective-,y. A short vertical segment 37 is formed along canti- .ever Cimmediately overlying that point on support A where the cantilever isembedded at its supported and. As is apparent from FIGS. 1 and 2, thecommon plane of zigzags 14 is normal to the common plane of zigzags 16.

Having described the supported cantilever, the sinos- )idal vibratorymotion which is supported mass describes :an now be set forth. Forsimplicity in understanding, such motion will only be described withrespect to lower Ligzags 14.

Considering FIG. 3, mass B is shown set in motion tlong a plane which isnormal to that plane of lower planar zigzags 14. The apparatus can beconsidered for :heoretical purposes to be a mass B oscillating about agoint in space. The movement of the mass can be conveniently describedusing Cartesian coordinates taken along a plane normal to the view ofFIG. 3. The viaratory motion of the mass is described by equating theiynarnic forces acting on the mass to the static forces which act on themass. Assuming that vibration in the node shown in FIG. 3 occurs alongthe X axis (with X=O being equivalent to the rest position) the dynamicforces acting on the mass can be obtained from Newtons second law ofmotion:

1 m an where F equals the force acting on the mass along the X axis, Mis the mass of ball B, A is acceleration (expressed as the secondderivative of X axis movement in :he second part of the equation), and tis time.

The restoring forces acting on the mass can be ob- :ained from Hookeslaw and are equal to:

where K is the spring constant of the cantilever in the X axis (whichspring constant is assumed to be substantially linear), and F is thestatic forces acting on the mass.

Since F and F are the only forces acting on the mass in the X direction,the two may be equated:

and integrated to obtain the periodic motion of the mass as a functionof time t to give the equation:

X=A sin H-Qr) where A is the constant for the initial amplitude ofvibration along the X axis and Q is the constant for the angularposition of the mass at time zero. It is readily observed that thisfunction is essentially a sinusoidal motion.

The spring constant K of the cantilever in its vibration about the Xaxis will include torsion imparted to substantially horizontal segments20, 21 and 22. These horizontal segments will tend to elasticallyro'tate rather than elastically bend as the mass vibrates from oneextreme position relative to support A to the opposed extreme positionrelative to support A in the plane of FIG. 3.

Referring to FIG. 4, vibration of the mass B in the plane of firstplanar zigzags 14 is illustrated. This motion occurs along the Y axis(with Y=O being the rest position). Similar to the vibration formulapreviously derived, the vibrational motion of the mass in the Y axiswill equal:

{K y S111 W -i-Qz) where A is the original amplitude in the Y direction,K is the spring constant in the Y direction and Q is a constant for theangular displacement at time zero. It will be observed that lower zigzag14 here is subjected to bending forces rather than the torsion forcespreviously illustrated in FIG. 3.

It has been found that the torsion spring constant K for a wire can besubstantially less than the bending spring constant K Accordingly, theforces experienced by the vibratory mass in the vibrational mode alongthe X axis of FIG. 3 are substantially less than those shown for thevibratory motion along the Y axis shown in FIG. 4. Typically, when themass is vibrated normal to the plane of first zigzags 14, the mass Bwill easily remain in such motion along its X axis. When the mass isvibrated parallel to the plane of the lower zigzag or along its Y axis,it will only remain in such motion when the mass is set in motionprecisely along the Y axis. Any vibrational movement imparted to themass B slightly deviated from the Y axis will cause the mass B to seek avibrational path of least resistance including motion along the X axisand consequently will result in a vibrational motion tracing Lissajousfigures. This vibrational motion can best be described with reference toFIGS. 5a through 5d.

FIG. 5a assumes that the mass has been set in motion along path Pinclined at a small counterclockwise acute angle relative to the axis Y.In such a vibratory motion the mass B will oscillate back and forthalong path P for a short period of time and then seek a vibratory path P(shown in FIG. 5b) which includes vibration in the X axis.

As shown in FIG. 5b when vibratory movement including substantialmovement in the X axis is attained, mass B will tend to follow asubstantially circular oscillating path counterclockwise about supportA. This substantially circular path will continue for a short period oftime; thereafter, the forces acting on the mass will tend to return themass to a vibrational path P substantially parallel to the Y axis shownin FIG. 5c.

With reference to FIG. 5c, the mass will approach a vibratory path Pwhich adjoins the Y axis and is inclined clockwise to the Y axis at asmall acute angle. Mass B will oscillate along this path for a shortperiod of time and then seek a path R, which again includes a vibratorymotion in the X axis. Similar to path P path P will be circular but willgenerally be substantially clockwise with respect to support A. The masswill vibrate along path R; for a short period of time finally seeking avibratory motion which then includes a path substantially similar topath P originally illustrated in FIG. 5a. The vibratory motion of FIGS.5a through 5d will then be sequentially repeated.

The mass along its vibratory path will thus describe a plainly visibleLissajour figure. This figure will include sequential vibrationssubstantially parallel to the Y axis, vibrations circular about base Ain a first direction, vibrations again substantially parallel to the Yaxis, and finally an opposed circular path about base A. This patternwill be repeated so long as the forces of friction do not damp andarrest the motions of the mass B.

The vibratory motion of the mass has thus far been described withreference to the lower zigzag 14. Upper zigzag 16 complements suchmotion. As is apparent, this upper zigzag 16 has its stable mode in theY axis, and its unstable mode in the X axis. While the upper zigzag 16is not necessary for practice of this invention, it does tend to impartadditional independent restoring forces to the moving mass andcomplements the lower zigzag so as to give the vibrational toy asymmetrical appearance.

The specific form of bends in cantilever C used to produce the differingvibrational constants K (torsion) and K (bending) are not critical. Anybend which disposes a segment of the wire in a substantially horizontaldisposition relative to the vertical axis of the cantilever C will tendto produce the differing torsion and bending elastic constants necessaryfor the generation of the described Lissajous patterns of motion. Thebends here shown, however, are preferred to accent such motion.

It is possible to construct a cantilever C which is vertical and withoutzigzags to accomplish the described vibratory motion. Such a cantileverC is illustrated in the cross section of FIG. 6.

With reference to FIG. 6, a cantilever cross section is illustratedhaving a circular and hollow interior 40 surrounded by an ellipticallyshaped resilient mass 42. Since the resilient mass 42 has the bulk ofits material disposed closer to the X axis than to the Y axis, forcesacting upon a vibrating mass along the Y axis will tend to be less thanforces acting upon a'mass vibrating in the X axis. Unfortunately,resilient elements constructed in the configuration shown in FIG. 6 arevery difficult to make; it is immediately realized that simple bendingof circular tinned music wire according to the Zigzags of the presentinvention is a greatly simplified construction.

It is known that cantilevers undergoing elastic deformation experiencetheir greatest elastic deformation adjoining the supported end of thecantilever. Accordingly, it is preferred to dispose a planar zigzagimmediately adjoining the support A. In this position, the differingelastic constants K, of torsion and K of pure bending are accented. Thisaccentuation results in creating the maximum differential between theresultant elastic forces in the X and Y axis.

As is apparent, vibration of mass B can also be attained by moving themass upwardly or downwardly with re spect to support A. This vibrationcompresses zigzags 14 and 16 and results in an additional vibrationalmotion imposed upon the mass in the Z axis (shown in FIG. 2).Unfortunately, in the embodiment of the invention shown here, vibrationsin the Z axis are much more rapid than those occurring in the X and Yaxis. These rapid vibrations make the visual observance of any Lissajouspattern occurring simultaneously in the X, Y and Z axis difiicult.

It has been found, however, that if cantilever C is horizontallydisposed, the vibratory motions along the Z axis can be made to approachthe frequency of the vibrations occurring in the X and Y axis. Thislatter construction permits three-dimensional Lissajous figures to betraced by the vibratory motion of the mass B. In such an embodiment itis preferred that the wire of cantilever C be reduced in size andresiliency so that the periodic motions of the mass become much moregradual.

What is claimed is:

1. In a toy for providing vibrational movement of a mass about a base anelastic cantilever having a fixed end supported from said base and afree end mounted for swaying deflection about its fixed end; a massmounted to the free end of the cantilever for vibratory motion to andfrom a position where said mass is at rest on the end of saidcantilever; said cantilever including an uncoiled wire formed with atleast one co-planar zig-zag.

2. The invention of claim 1 wherein said cantilever is supportedvertically with said mass mounted to the upper end of said cantilever.

3. The invention of claim 1 wherein said elastic cantilever includes atleast one segment of said wire disposed substantially normal to thelongitudinal axis of said cantilever.

4. The invention of claim 1 wherein said co-planar Zig-zag is disposedadjacent the supported end of said cantilever.

5. The invention of claim 1 wherein said coplanar zigzag is within aplane that includes the longitudinal axis of said cantilever.

6. The invention of claim 5 wherein said cantilever includes a pluralityof said zig-zag segments, said co-planar zig-zag segments each beingformed in different planes.

7. In a toy for providing vibrational movement of a mass about asupport, the combination with said support and mass comprising: anelastic cantilever having a fixed end and a free end attached to saidsupport at one end and to said mass at the opposite end for supportingsaid mass at a rest position relative to said support; said elasticcantilever being a single uncoiled wire formed with at least oneco-planar zig-zag.

8. The invention of claim 7 wherein said co-planar zigzag includes asegment of said wire substantially normal to the axis of said cantileverand disposed adjacent the supported end of said cantilever.

9. The invention of claim 7 and wherein said elastic cantilever furtherincludes a second co-planar zig-zag disposed in a plane including alongitudinal axis of said cantilever and intersecting the plane of afirst co-planar zigzag.

10. The invention of claim 7 wherein the plane of said second co-planarzig-zag is perpendicular to the plane of a first co-planar zig-zag.

References Cited UNITED STATES PATENTS 8/1932 Worthington 33-27 2/1962Zinnow 4632 X US. Cl. X.R. 33-27; 35--l9, 30

